is trivially satisfied, as LH is a degenerate cartesian product with only one
component.
Indeed, the idea of the state of a system as a function assigning results to observation ties in nicely with the abstraction from Hilbert spaces given byF. It remains to be seen whether we can exploit the deconstruct-into-subsystems mechanism of Correlation Models to obtain a tensor operator.
5.4
Tensor on Correlation Models
We have the following wishlist for a tensor functor on Correlation Models: 1. it must be of type D×D→D, whereDis CM[0,1] or a subcategory 2. it must match the tensor in FdHil
3. givenAH andAV, the compound correlation model must be a set of functions of typeLH ×LV →[0,1]
4. the partial trace inFdHilmatches the subsystem operation inCM[0,1] The third condition is what is required to apply the deconstruct-into- subsystems mechanism of correlation models.
At first sight one might be tempted to define the set AH ⊗ AV as
{P
ibi(siH, siV)|bi ∈ [0,1],Pibi = 1, siH ∈ P ure(AH), siV ∈ P ure(AV)},
where P
ibi(siH, sVi ) :LH ×LV →[0,1] is the function defined as X i bi(siH, siV)(aH, aV) = X i bisiH(aH)siV(aV)
If we follow this path though, in proving the matching between AH ⊗AV
andAHN
V we are forced to send the (probability function associated to the)
density operator of an entangled states such asβ00 to a mixture of separable states, like P
ibi(siH, siV). But this essentially would mean that entangled
states are mixed states, contradicting the fact that some entangled states are pure.
The reason why we cannot write the density operator of an entangled state as a mixture of density operators of separable states is that the former contains matrices that are not density operators.
Example
Consider for example the density operator of the Bell stateβ00:
|00i+|11i √ 2 h00|+h11| √ 2 = = |00i h00|+|00i h11|+|11i h00|+|11i h11| 2 = |0i h0| N |0i h0|+|0i h1|N |0i h1|+|1i h0|N |1i h0|+|1i h1|N |1i h1| 2
The matrix |00i h11|, for example, is not of trace 1, and thus is not a density operator.
This points to a more general problem. In general the density operators in a bipartite system cannot be written in terms of the density operators of the two component systems, we need also additional terms, as highlighted in the example. Such terms have shape|αi hβ|, where |αi,|βi are two orthogonal vectors. So it seems that to recover these terms we need to know that the functions that we see as states are density operators over some Hilbert space. If we do not, then the states of the component systems do not contain enough information to obtain the states of the bipartite system. We have however no proof for these speculations; they will be subject of future work.
Conclusions and future work
We have seen how from the categoryFdHil and the functorS we can obtain a class of Modal Logic frames forLQP andLQPn. This constitutes the link between the two research programs that we have considered.
Our second case study, the functor F, has highlighted the possibility to obtain a richer semantics. We have designed a logic for such class of Modal Logic frames, and proven that it is an improvement with respect to LQP
andLQPn, in the sense that it has more expressive power and it contains all the (translations of) the theorems of LQP and LQPn.
On a more general level, we have moved the first steps in the study of
LSC logics. We have analyzed three examples,DLT,S4 and Hybrid Logic, showing how different languages, different categories and different functors can be used to characterize certain classes of (C, U)-frames.
These results prompt two groups of questions, one related to the main topic of this thesis and one at the intersection of Category Theory and Modal Logic. We start with the former group.
1. A first problem concerns the design of a proof system of ΛFP robn Γn
F . The application of this proof system should be the correctness proof of a quantum protocol where probability plays an essential role.
2. It would be interesting to explore the connection between ΛFP robn Γn
F and other logics that have been proposed in the area, such as the calculus for dagger compact closed categories with biproducts presented in [5] and the logics proposed in [10], [9] and [8].
3. The two problems mentioned in Chapter 5 are still undecided. In par- ticular, we would like to understand which are the conditions needed in order to have a tensor operator and, conversely, under which conditions it is impossible to have it.
4. The different functors fromFdHiltoReland the corresponding classes of Modal Logic frames represent different possible abstractions from Hilbert spaces; it would be interesting to study the connection with other abstractions from Hilbert spaces, as for example the Chu spaces described in [2] or the coalgebras in [1].
We now turn to the second, more abstract, group of questions.
1. The functors from a category Cto Rel constitute a category called RelC, having as morphisms natural transformations. Therefore we can investigate the connection between the properties of this category and the properties of the corresponding logics. For example:
• How does the existence of a natural transformation between two functors riverberate on the corresponding logics? Note that the transformation of the Modal Logic frames given byF into Modal Logic frames given byS (Lemma Chapter 4) was allowed by a natural transformation.
• Can we characterize the logics arising from initial and terminal functors?
• We know that most of the categorical structure of Rel can be lifted toRelC via componentwise definitions. But we also know that Rel is a degenerate dagger compact closed category with byproducts. How does this affect the class of logics arising from the functors inRelC?
• If we restrict our attention to the category of functorsSetC, i.e. the category of functors from a small cateogry intoSet, we can see thatSetC is a topos.1 It would be interesting to investigate the interplay between Topos Theory and our procedure to obtain Modal Logic frames.
2. The main contemporary field at the intersection of Modal Logic and Category Theory is Coalgebra. We have briefly mentioned coalgebras, or rather the “coalgebraic approach”, in Chapter 5, when we were discussing the characterization of the tensor. On a more general level, we are interested in understanding the link of Coalgebra with our procedure.
We will pursue these lines of research in future work.
1
Acknowledgements
I would like to thank Alexandru Baltag, for insightful comments and discus- sions, Ronald de Wolf and Christian Schaffner, for their patience in listening to my presentations, and Kohei Kishida, always kind and helpful.
On a personal level, a big hug goes to my parents, for their unquestioning support, and to my girlfriend, who endured my grumpyness throughout these months.
Appendix A
Basic Notions
In this Appendix we review the basic definitions and facts used in the thesis.
A.1
Category Theory
The reference for this section is [19].
Definition 75. A category Cis a structure consisting of • a collection ofobjects, denoted C0
• a collection ofmorphisms, or arrows, denoted C1
• two operationsdomandcod, “domain” and “codomain” (sometimes called “source” and “target”), assigning an object to each morphism We write f : A → B to mean that f is a morphism with domain A and codomainB.
• for each object A in C0, there is an arrow IdA : A → A, called the identity of A
• for each pair of morphismsf, g inC1 such that cod(f) =dom(g) there is an arrowg◦f called the composite1
To be a category such structure is required to satisfy the following axioms: 1. Associativity: for all f, g, h in C1 with the correct configuration of
domains and codomains,
(f◦g)◦h=f ◦(g◦h)
1
Note that the order of the morphisms in the composite is reversed with respect to the order of “application” of the morphisms.
2. Identity Axioms: for allf :A→B in C1,
f◦IdA=f =IdB◦f
Definition 76. Call Setthe category having sets as objects and functions as morphisms. The identity morphisms are the identity functions, the composition is the composition of functions.
Definition 77. Call Relthe category having sets as objects and relations as morphisms. The identity morphisms are the identity relations, the com- position is the composition of relations.
Definition 78. A morphismf :A→B in a categoryCis anisomorphismif there is a morphismf−1 :B→A, called theinverse, such thatf◦f−1 =IdB
andf−1◦f =IdA. We writef :A'B to indicate thatf is an isomorphism. Definition 79. Given a categoryC,A, B, C, D inC0 and f, g, f0, g0 inC1, a diagram such as A B C D f g0 f0 g is said tocommute ifg◦f =f0◦g0.
Note that a commuting diagram can represent different mathematical statements depending on the specific category under consideration. If for exampleCisSet, the commutation of the diagram above means that,for all x∈A,g◦f(x) =f0◦g0(x).
Definition 80. An objectA in a categoryCis initial if for every objectB
inC0 there is a unique arrowA→B. An object Ais terminal (sometimes also calledfinal) if for every object B there is a unique arrowB →A. Definition 81. Given objectsA, Bin a categoryC, theirproductis an object
A×B equipped with two morphismsp1:A×B →A andp2:A×B→B such that for all objectsCin C0 and morphismsf1:C →Aandf2 :C→B there exists a unique morphismg :C → A×B such that the triangles in the following diagram commute
A.1. CATEGORY THEORY 91 C A A×B B g f1 f2 p1 p2
A category has products if for every pair of objects their product exists in the category.
Definition 82. Given objects A, B in a category C, theircoproduct is an objectA+Bequipped with two morphismsi1:A→A+Bandi2 :B →A+B such that for all objectsC inC0 and morphismsf1 :A→C andf2:B →C there exists a unique morphismg:A+B →C such that the triangles in the following diagram commute
A A+B B C g i1 f1 i2 f2
A category has coproducts if for every pair of objects their coproduct exists in the category. Notice that the coproduct is the dual notion of the product: it is obtained by reversing all the arrows in the definition of product. Definition 83. Given two categoriesCand D, a functor G fromCtoD, written G:C→D, is a pair of functions
1. G0 from C0 toD0 2. G1 from C1 toD1 such that
• for allf inC1,dom(G1(f)) =G0(dom(f)) andcod(G1(f)) =G0(cod(f)); a functorpreserves domains and codomains
• for all AinC0,G1(IdA) =IdG0(A); a functor preserves identities
• for all f, g in C1 G1(f ◦ g) = G1(f) ◦ G1(g); a functor preserves
It is customary to abuse the notation and drop the subscript from the functor. When specifying a functorGwe write
A7→. . . f :A→B 7→. . .
where the first line describes the action ofG0 and the second line the action of G1. An ‘endofunctor is a functor going from a category to itself. A
contravariant functor is a functor fromCtoDis a functor of type Cop→D.
Proposition 20. Functors preserve commuting diagrams: if G :C → D
and A B C D f g0 f0 g
is a commuting diagram inCthen
G(A) G(B) G(C) G(D) G(f) G(g0) G(f0) G(g)
is a commuting diagram inD. As a consequence, functors preserve isomor- phisms.
Definition 84. Call Cat the category having categories as objects and functors as morphisms.
Definition 85. Given two categoriesCandD, theproduct category C×D is composed as follows
• the objects are pairs of objects (C, D), whereC is inC0 and Din D0 • the morphisms are pairs of morphisms (f, g), where f is in C1 andg
inD1
the identity is the pair of identities and the composite is the pairwise com- posite. The categoryC×D is the product of CandD in the categoryCat in the sense of Definition 81.
A.2. QUANTUM MECHANICS 93